This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem cbviotaw

Description: Change bound variables in a description binder. Version of cbviota with a disjoint variable condition, which does not require ax-13 . (Contributed by Andrew Salmon, 1-Aug-2011) Avoid ax-13 . (Revised by GG, 26-Jan-2024)

Ref Expression
Hypotheses cbviotaw.1 
|- ( x = y -> ( ph <-> ps ) )
cbviotaw.2 
|- F/ y ph
cbviotaw.3 
|- F/ x ps
Assertion cbviotaw 
|- ( iota x ph ) = ( iota y ps )

Proof

Step Hyp Ref Expression
1 cbviotaw.1 
 |-  ( x = y -> ( ph <-> ps ) )
2 cbviotaw.2 
 |-  F/ y ph
3 cbviotaw.3 
 |-  F/ x ps
4 nfv 
 |-  F/ z ( ph <-> x = w )
5 nfs1v 
 |-  F/ x [ z / x ] ph
6 nfv 
 |-  F/ x z = w
7 5 6 nfbi 
 |-  F/ x ( [ z / x ] ph <-> z = w )
8 sbequ12 
 |-  ( x = z -> ( ph <-> [ z / x ] ph ) )
9 equequ1 
 |-  ( x = z -> ( x = w <-> z = w ) )
10 8 9 bibi12d 
 |-  ( x = z -> ( ( ph <-> x = w ) <-> ( [ z / x ] ph <-> z = w ) ) )
11 4 7 10 cbvalv1 
 |-  ( A. x ( ph <-> x = w ) <-> A. z ( [ z / x ] ph <-> z = w ) )
12 2 nfsbv 
 |-  F/ y [ z / x ] ph
13 nfv 
 |-  F/ y z = w
14 12 13 nfbi 
 |-  F/ y ( [ z / x ] ph <-> z = w )
15 nfv 
 |-  F/ z ( ps <-> y = w )
16 3 1 sbhypf 
 |-  ( z = y -> ( [ z / x ] ph <-> ps ) )
17 equequ1 
 |-  ( z = y -> ( z = w <-> y = w ) )
18 16 17 bibi12d 
 |-  ( z = y -> ( ( [ z / x ] ph <-> z = w ) <-> ( ps <-> y = w ) ) )
19 14 15 18 cbvalv1 
 |-  ( A. z ( [ z / x ] ph <-> z = w ) <-> A. y ( ps <-> y = w ) )
20 11 19 bitri 
 |-  ( A. x ( ph <-> x = w ) <-> A. y ( ps <-> y = w ) )
21 20 abbii 
 |-  { w | A. x ( ph <-> x = w ) } = { w | A. y ( ps <-> y = w ) }
22 21 unieqi 
 |-  U. { w | A. x ( ph <-> x = w ) } = U. { w | A. y ( ps <-> y = w ) }
23 dfiota2 
 |-  ( iota x ph ) = U. { w | A. x ( ph <-> x = w ) }
24 dfiota2 
 |-  ( iota y ps ) = U. { w | A. y ( ps <-> y = w ) }
25 22 23 24 3eqtr4i 
 |-  ( iota x ph ) = ( iota y ps )